In this paper, we investigate the growth of solutions of the differential equation \[ f''+A(z)\expeft\{\frac{a}{(z_{0}-z)^n}\right\}f'+B(z)\expeft\{\frac{b}{(z_{0}-z)^n}\right\}f=0, \] where $A(z)$, $B(z)$ are analytic functions in the closed complex plane except at $z_{0}$ and $a,b$ are complex constants such that $ab\neq 0$ and $a=cb$, $c>1$. Another case has been studied for higher order linear differential equations with analytic coefficients having the same order near a finite singular point.